The demand and supply functions for a good are P d = 50 2Q d and P s = 14 + 4Q s respectively, where P and Q denote price and quantity. a. Find the equilibrium price and quantity b. Find the…

Model Answer: Basic Mathematics (MTH204) – TU Board 2023

Question: The demand and supply functions for a good are $$P_d=50-2Q_d\text{ and }{P}_s=14+4Q_s$$ respectively, where $P$ and $Q$ denote price and quantity.

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(a) Find the equilibrium price and quantity

Step 1: Understand Equilibrium Condition

At equilibrium, the quantity demanded ($Q_d$) equals the quantity supplied ($Q_s$), and the price demanded ($P_d$) equals the price supplied ($P_s$). Thus: $$Q_d=Q_s=Q^{\ast }$$ $$P_d=P_s=P^{\ast }$$

Step 2: Set Demand = Supply and Solve for $Q^{\ast }$

Given: $$P_d=50-2Q_d$$ $$P_s=14+4Q_s$$

At equilibrium: $$50-2Q^{\ast }=14+4Q^{\ast }$$

Bring like terms together: $$50-14=4Q^{\ast }+2Q^{\ast }$$ $$36=6Q^{\ast }$$ $$Q^{\ast }=\frac{36}{6}=6$$

Step 3: Find Equilibrium Price ($P^{\ast }$)

Substitute $Q^{\ast }=6$ into either $P_d$ or $P_s$. Using $P_d$: $$P^{\ast }=50-2(6)=50-12=38$$

Verification using $P_s$: $$P^{\ast }=14+4(6)=14+24=38$$ Both give the same result, confirming equilibrium.

Final Answer:

• Equilibrium Quantity ($Q^{\ast }$) = 6 units
• Equilibrium Price ($P^{\ast }$) = ₹38

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(b) Find the consumer’s surplus and producer’s surplus at equilibrium

Consumer’s Surplus (CS)

Consumer’s surplus is the area between the demand curve and the equilibrium price, up to $Q^{\ast }$. It is given by: $$CS=\frac{1}{2}\times (P_{\text{max}}-P^{\ast })\times Q^{\ast }$$ Where:

• $P_{\text{max}}$ is the price intercept of the demand curve (when $Q_d=0$).
• From $P_d=50-2Q_d$, when $Q_d=0$, $P_{\text{max}}=50$.

Substitute values: $$CS=\frac{1}{2}\times (50-38)\times 6$$ $$CS=\frac{1}{2}\times 12\times 6$$ $$CS=\frac{1}{2}\times 72=36$$

Producer’s Surplus (PS)

Producer’s surplus is the area between the supply curve and the equilibrium price, up to $Q^{\ast }$. It is given by: $$PS=\frac{1}{2}\times (P^{\ast }-P_{\text{min}})\times Q^{\ast }$$ Where:

• $P_{\text{min}}$ is the price intercept of the supply curve (when $Q_s=0$).
• From $P_s=14+4Q_s$, when $Q_s=0$, $P_{\text{min}}=14$.

Substitute values: $$PS=\frac{1}{2}\times (38-14)\times 6$$ $$PS=\frac{1}{2}\times 24\times 6$$ $$PS=\frac{1}{2}\times 144=72$$

Final Answer:

• Consumer’s Surplus (CS) = ₹36
• Producer’s Surplus (PS) = ₹72

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(c) Find total surplus

Total surplus is the sum of consumer’s surplus and producer’s surplus at equilibrium: $$\text{Total Surplus}=CS+PS$$ $$\text{Total Surplus}=36+72=108$$

Final Answer:

• Total Surplus = ₹108

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Graphical Representation (ASCII)

Price (P)
  ^
50 |               Demand Curve
   |              /
   |             /
   |            /
38 |-----------*----------- Equilibrium Point (P*, Q*)
   |           / \
   |          /   \
   |         /     \
14 |        /       Supply Curve
   |_______/____________________>
      0     6       Q (Quantity)

• Demand curve intercepts at (0,50) and (25,0).
• Supply curve intercepts at (0,14) and (-3.5,0).
• Equilibrium at (6,38).

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Key Notes for Examiners:

1. Equilibrium Calculation:

   • Correctly set $Q_d=Q_s$ and solved for $Q^{\ast }$.
   • Verified $P^{\ast }$ using both demand and supply equations.

2. Surplus Calculations:

   • Used the triangle area formula for both CS and PS.
   • Identified correct intercepts ($P_{\text{max}}=50$, $P_{\text{min}}=14$).

3. Total Surplus:

   • Summed CS and PS accurately.

4. Clarity:

   • Step-by-step derivation with units (₹) for economic interpretation.
   • ASCII graph included for visual aid (though not required, it reinforces understanding).

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End of Answer.